Alan Rickman had cello lessons, and handles the right (bowing) hand, but the left hand is provided by a real cellist standing behind him with his arm through Alan Rickman's armpit. Juliet Stevenson does play her piano part however.
Stevenson actually plays the piano in this film. While Rickman needs a little help with the cello Stevenson did all of her own scenes with the piano. Either she had played before or was a quick learner since she didn't need a double to stand in and play for her.